library(mosaic)
library(tidyverse)
library(pander)
library(DT)
library(ggrepel)
library(plotly)
library(dplyr)
library(ggplot2)
library(maps)
library(tmap)
library(leaflet)
library(htmltools)
library(car)
library(mosaicData)
library(ResourceSelection)
library(reshape2)
library(RColorBrewer)
library(scatterplot3d)
library(readr)

ExamsTutor <- read_csv("C:/Users/paige/OneDrive/Documents/Fall Semester 2024/MATH 325/Statistics-Notebook-master/Data/ExamsTutor.csv")

Background


Brigham Young University-Idaho offers free tutoring for all students in various subjects through personal tutors, group tutoring, or drop-in labs. The introductory math classes (Math 100A, 100B, and 101) at BYU-Idaho collaborate with the university’s Math Study Center by assigning “Tutor Visits.” These assignments require students to visit the free Math Study Center for help with challenge questions. Completing these tutor visits contributes to the Tutoring category, which accounts for 10% of the final grade. The purpose of these assignments is twofold: to encourage students to utilize available resources and to enhance their learning. Ultimately, tutoring aims to improve performance in other areas, such as assignments and tests. In this study, we will address the following question:

Does the amount of tutoring a student receives correlate with their chapter exam scores?


To see the data, click the tab below

Hide Data

Show Data

In this study, we will be using students from Math 100B (Beginning Algebra). The rows represents a student, and the columns show the students’ Tutoring Final Score and their Chapter Exam Final Score.

Tutoring Final Scores are calculated by how many Tutor Visit assignments the student completed and then averaged to produce a final score, and Chapter Exam Final Scores are given by averaging their Chapter Exam Scores throughout the semester into an overall Exam Final Score.

datatable(ExamsTutor, options(list=c(3,10,30)))

Hypothesis


In this study, the following model show the relationship between a student’s Final Chapter Exam Score through a linear regression.

\[\underbrace{Y_i}_\text{Exam Final Scores} = \overbrace{\beta_0}^\text{Y- intercept} + \overbrace{\beta_1}^\text{Slope} \underbrace{X_i}_\text{Final Tutoring Scores} + \epsilon_i \space where \space \epsilon_i \sim N(0,\sigma^2)\]

In linear regression, the key points of interest are the y-intercept and the slope. The y-intercept isn’t particularly helpful in this case, as it only tells us the average final exam score for students who don’t receive tutoring. Instead, we’ll focus on the slope, which reveals how exam scores change relative to the percentage of tutoring a student receives. Our hypotheses will be based on this relationship:

\[H_0 : \beta_1 = 0\] \[H_a :\beta_1 \neq 0\]


Additionally, our level of significance will be:

\[\alpha = 0.05\]

Analysis


A scatter plot of Tutoring Final Scores vs. Exam Final Scores below shows a moderately positive relationship. The correlation coefficient of 0.6323 indicates a moderate linear association between these variables.

Hover over each dot to see specific scores

TTExams <- ggplot(ExamsTutor, aes(x=`Tutoring Final Score`, y= `Exams Final Score`)) +
  geom_point(size=1.5, color = "darkolivegreen", alpha =0.5) +
  geom_smooth(method="lm", formula= y~x, se=FALSE, size= 0.5, color="darkgreen")+
  labs(title="BYU-Idaho's 100B Students Tutoring Effects on Chapter Tests") +
  theme_minimal()

ggplotly(TTExams)
Correlation (r)
cor(ExamsTutor$`Tutoring Final Score`,ExamsTutor$`Exams Final Score`) %>%
  pander()

0.6323


To further confirm our findings, we will now conduct a Simple Linear Regression on the data set.


Simple Linear Regression Model


To further confirm our findings, we will now conduct a Simple Linear Regression on the data set. The results are shown below:

lmTTExams <- lm(`Exams Final Score` ~ `Tutoring Final Score`, data= ExamsTutor)

pander(summary(lmTTExams))
  Estimate Std. Error t value Pr(>|t|)
(Intercept) 57.21 4.573 12.51 2.07e-19
Tutoring Final Score 0.3474 0.05125 6.779 3.339e-09
Fitting linear model: Exams Final Score ~ Tutoring Final Score
Observations Residual Std. Error \(R^2\) Adjusted \(R^2\)
71 9.972 0.3998 0.3911



Using the results of the Linear Regression, answers our question of how the amount of tutoring a student receives correlates with their chapter exam scores with the following equation:

\[\underbrace{\hat{Y_i}}_\text{Mean Exam Final Score} = 57.21 + 0.3474 \underbrace{X_i}_\text{Tutoring Final Score}\]

Due to the size of our p-value being \(3.339e^{-9}\), we have significant evidence of a meaningful relationship between tutoring and exam scores. Based on our model, a student’s average chapter test final score would increase by 0.3474% for every 1 unit increase of tutoring that the student gets done.

\[P- value = 3.339e^{-9} < \alpha\]


While the model offers us great insight into our data, we must first check if we can trust the findings of this data by validating its appropriateness.


Go to the next tab to see how we check for appropriateness

Checking Simple Linear Regression Assumptions


There are 5 assumptions in our Linear Regression model:

  1. Constant Variance

  2. Independent Errors

  3. Normal Errors

  4. Fixed X Values

  5. Linear Relation


The following three diagnostic plots will help to identify if our assumptions are violated or not.

The 4th assumption, Fixed X Values, can’t be tested with a diagnostic plot. However, since the data was collected directly from the grade book, we can reasonably assume accurate and precise measurement of the X variable.

par(mfrow=c(1,3))

plot(lmTTExams, which=1)

qqPlot(lmTTExams$residuals, main="Q-Q Plot", col="darkolivegreen", col.lines="darkgreen",pch= 19, id=FALSE)

plot(lmTTExams$residuals, ylab= "Residuals", main="Residuals vs Order")

The Residuals versus Fitted Values plot assesses Linear Relation and Constant Variance.

Despite a slight linear pattern at the end, the data shows constant variance and linearity due to randomly scattered residuals. Thus, our 1st and 5th assumptions are NOT violated.
The Q-Q Residuals plot assesses Normal Errors. Outliers at the top and bottom indicate non-normality of residuals, violating our 3rd assumption.
The Residuals versus Order plot checks for Independent Errors. With no clear pattern, we can assume residual independence, thus not violating our 2nd assumption.



While one assumption is violated due to outliers in the Q-Q Residuals plot, the impact on our results is not drastic. However, interpretations should still be made cautiously, considering the slight skewness in the data.


Interpretation


Our Simple Linear Regression analysis and accompanying graphics revealed that there is a correlation between the amount of tutoring a student receives and their chapter exam scores.

The scatter plot demonstrated a moderate positive relationship between tutoring and chapter exam scores. Our model indicated that for every 1 unit increase in tutoring participation, a student’s average chapter exam final score would increase 0.3474 as well. In other words, more tutoring correlated with higher average chapter exam scores. Focusing on the slope, we obtained a p-value of \(3.339e^{-9}\), indicating a statistically significant relationship between the two variables. These findings suggest that students aiming to enhance their knowledge and test performance should strongly consider engaging in tutoring.

However, it’s crucial to note that since the data set violated one of the Simple Linear Regression assumptions, these interpretations and recommendations should be viewed with caution. While these findings may not be conclusive, seeking tutoring for any subject remains a valuable pursuit.

Sources